Juan Carlos Bustamante
We show that the graded commutative ring structure of the Hochschild cohomology HH*(A) is trivial in case A is a triangular quadratic string algebra. Moreover, in case A isgentle, the Lie algebra structure on HH*(A) is also trivial.
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In this paper we define a new cohomology for multiplicative Hom-associative algebras, which generalize Hochschild cohomology and fits with deformations of Hom-associative algebras including the structure map $α$. It is a generalization of the known Hochschild-type cohomology for Hom-associative algebras defined by Amar, Ejbehi and Makhlouf. Moreover, we provide various observations and similarly a new cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras given by Dekkar and Makhlouf.
For a finite-dimensional Frobenius $k$-algebra $R$ with the Nakayama automorphism $ν$ we define an algebra ${\rm HH}^*(R)^{ν\uparrow}$. If the order of $ν$ is not divisible by the characteristic of $k$, this algebra is isomorphic to the Hochschild cohomology algebra of $R$.. We prove that this algebra is a BV-algebra. We use this fact to calculate the Gerstenhaber algebra structure and BV-structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type $D_n$.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
The aim of this paper is twofold. In the first part, we consider twisted Rota-Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an $L_\infty$-algebra whose Maurer-Cartan elements are given by twisted Rota-Baxter operators. This leads to cohomology associated to a twisted Rota-Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota-Baxter operators by means of the above-de…
This paper studies averaging algebras, say, associative algebras endowed with averaging operators. We develop a cohomology theory for averaging algebras and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions of averaging algebras. We make explicit the $L_\infty$-algebra structure over the cochain complex defining cohomology groups and introduce the notion of homotopy averaging algebras as Maurer-Cartan elements of this $L_\infty$-algebra.
For algebras with radical square zero, we give a combinatorial description to the Tate-Hochschild cohomology. We compute the Gerstenhaber algebra structure on the Tate-Hochschild cohomology for some classes of such algebras.
This document is supplementary material for the author's paper \emph{The cup product on hochschild cohomology for localizations of filtered koszul algebras}, arXiv:1304.0527. We give a detailed computation of the Hochschild cohomology algebra for the Down-Up algebra with parameters $(0,1,0)$.
This paper is devoted to studying deformation, cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights. First we give the notion of a new representation of a Rota-Baxter pre-Lie algebra of arbitrary weight and define the cohomology theory of a Rota-Baxter pre-Lie algebra of arbitrary weight. Then we study formal deformations by lower degree cohomology groups. Finally, we classify abelian extensions of Rota-Baxter pre-Lie algebras of arbitrary weight using the second cohomology group, and classify skeletal Rota-Baxter pre-Lie 2-algebra of arbitrary weight using the third cohomolo…
On the cohomology of Lie algebras associated with graphs
本文研究图关联李代数的上同调，给出 Dani–Mainkar 亚阿贝尔李代数上同调的典范分解及显式公式，并将可解情形归约到该类李代数。
We consider cohomology of diagrams of algebras by Beck's approach, using comonads. We then apply this theory to computing the cohomology of $Ψ$-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra.
The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Al…
Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras
本文引入李代数匹配对的表示与上同调，证明其与李双代数上同调的态射关系，可用于无穷小形变与阿贝尔扩张研究，还定义了$L_\infty$-代数匹配对并构造双交叉积，揭示骨架$L_\infty$-代数匹配对与该上同调的关联。
The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT].
We show that there is an action of the symmetric group on the Hochschild cochain complex of a twisted group algebra with coefficients in a bimodule. This allows us to define the symmetric Hochschild cohomology of twisted group algebras, similarly to th construction of symmetric group cohomology due to Staic. We give explicit embeddings and connecting homomorphisms between the symmetric cohomology spaces and symmetric Hochschild cohomology of twisted group algebras.
We describe the Hochschild cohomology ring for one of the two self-injective algebras of tree class $E_6$ in terms of generators and relations.
In this paper we study the cyclic cohomology of certain x-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici x-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of x-Hopf algebras which generalizes the Connes-Moscovici characteristic map to x-Hopf algebras. This enables us to transfer the x-Hopf algebra cyclic cocycles to algebra cyclic cocycles.
We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra.
Let $(S,L)$ be a Lie-Rinehart algebra such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of $U$ with values on a $U$-bimodule $M$ and whose second page involves the Lie-Rinehart cohomology of the algebra and the Hochschild cohomology of $S$ with values on $M$. After giving a convenient description of the involved algebraic structures we use the spectral sequence to compute explicitly the Hochschild cohomology of the algebra of differential operators tangent to a central arran…
A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the…
We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of 'quantum' type in all but a few exceptional cases.
This paper generalizes Huang's cohomology theory of grading-restricted vertex algebras to meromorphic open-string vertex algebras (MOSVAs hereafter), which are noncommutative generalizations of grading-restricted vertex algebras introduced by Huang. The vertex operators for these algebras satisfy associativity but do not necessarily satisfy the commutativity. Moreover, the MOSVA and its bimodules considered in this paper do not necessarily have finite-dimensional homogeneous subspaces, though we do require that they have lower-bounded gradings. The construction and results in this paper will b…
We show that for some finite group block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial. Along the way we obtain methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras.
We give a basis for the Hochschild cohomology ring of tame Hecke algebras. We then show that the Hochschild cohomology ring modulo nilpotence is a finitely generated algebra of Krull dimension 2, and describe the support varieties of modules for these algebras. As a consequence we obtain the result that the Hochschild cohomology ring modulo nilpotence of a Hecke algebra has Krull dimension 1 if the algebra is of finite type and has Krull dimension 2 if the algebra is of tame type.
We define and study the cohomology theories associated to A-infinity algebras and cyclic A-infinity algebras equipped with an involution, generalising dihedral cohomology to the A-infinity context. Such algebras arise, for example, as unoriented versions of topological conformal field theories. It is well known that Hochschild cohomology and cyclic cohomology govern, in a precise sense, the deformation theory of A-infinity algebras and cyclic A-infinity algebras and we give analogous results for the deformation theory in the presence of an involution. We also briefly discuss generalisations of…
We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.
We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.
Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gers…
We prove that the cohomology algebra of a conilpotent Lie coalgebra is generated in degree 1 as an A-infinity algebra. By dualizing, the same is true about cohomology algebras of finite dimensional nilpotent Lie algebras. In the process, we provide a proof of the conilpotent version of the dual Poincare-Birkhoff-Witt theorem.
For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natura…
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